LC transfer of energy between the driving source and the circuit will be a maximum.

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1 The Q of oscillatos efeences: L.. Fotney Pinciples of Electonics: Analog and Digital, Hacout Bace Jovanovich 987, Chapte (AC Cicuits) H. J. Pain The Physics of Vibations and Waves, 5 th edition, Wiley 999, Chapte 3 (The foced oscillato). Peequisite: Cuents though inductances, capacitances and esistances nd yea lab expeiment, Depatment of Physics, Univesity of Toonto, Intoduction In the Peequisite expeiment, you studied LC cicuits with diffeent applied signals. A loop with capacitance and inductance exhibits an oscillatoy esponse to a distubance, due to the oscillating enegy exchange between the electic and magnetic fields of the cicuit elements. If a esisto is added, the oscillation will become damped. In this expeiment, the LC cicuit will be diven at esonance fequencyω, when the LC tansfe of enegy between the diving souce and the cicuit will be a maximum. LC cicuits at esonance. The tansfe function. Ohm s Law applied to a LC loop (Figue ) can be witten in complex notation (see Appendix fom the Peequisite) Figue LC cicuit fo esonance studies i( jω ) Z Ohm s Law: () - -

2 i( and ae complex instantaneous values of cuent and voltage, ω is the angula fequency ( ω πf ), Z is the complex impedance of the loop: Z + j ωl ωc () j is the complex numbe The voltage acoss the esisto fom Figue, as a esult of cuent i can be expessed as: v ( i( (3) Eliminating i( fom Equations () and (3) esults into: v ( + j( ωl / Equation (4) can be put into the geneal fom: (4) v ( H( (5) whee H( is called a tansfe function acoss the esisto, in the fequency domain. H ( + j( ωl / (6) H is an impedance atio, useful in descibing the esonance of a diven LC loop. Any complex numbe such as H( can be put in the fom: jθ ( ω ) H ( H A ( ω) e (7) whee H A (ω) is the eal amplitude (o magnitude) of the complex numbe: H A ( ω) (8) + ( ωl / (/ ωl and θ(ω) is the phase, defined as: θ ( ω) tan (9) The tansfe function detemines the phase and amplitude elationships between the voltage acoss esisto v ( (output) and the applied voltage (input). The tansfe function shows how a cicuit modifies the input signal in ceating the output. Mathematically speaking, the tansfe function completely descibes how the cicuit pocesses the input complex exponential to poduce the output complex exponential. We can chaacteize a cicuit function by examining the magnitude and phase of its tansfe function Fom Equations (8) and (9), note that amplitude H A (ω) is a maximum and the phase θ(ω ) is zeo when ( ω L / 0. This is called esonance. At esonance thee is maximum of - -

3 enegy tansfe between the diving souce and the cicuit. The esonant fequency of the cicuit is defined as: ω (0) LC The Q facto With efeence to a specific LC cicuit, the Q facto measues the stength of a esonance. Fo a seies LC loop, by definition: L L Q ω () C Moe fundamentally, the Q facto of a esonance is π times the stoed enegy divided by the enegy lost pe oscillation cycle. We can expess now the complex tansfe function with the Q facto, combining equations (6) and (): H ( () ω ω + jq ω ω A log-log plot of the magnitude of the tansfe function H ( as function of ω/ω fo diffeent values of Q would show the esonant behavio of a seies LC loop. Lage Q values coespond to naowe esonance cuves (Figue ): Figue esonant behavio of an LC loop - 3 -

4 Looking at Figue, we can define two fequencies ω and ω which satisfy: H ( jω, ) H ( jω ). ω and ω ae called half-powe fequencies. They allow ewiting the Q facto as: ω ω Q ω ω Δω V out Fo pactical applications, the atio (whee Vout V and V in ae voltage amplitude values Vin you measue on oscilloscope) can be used to detemine the esonance, instead of the tansfe function. A mechanical system at esonance (see efeences) An oscillato with a linea estoing foce and viscous damping γ obeys an equation of the fom: d x dx + γ + ω0 x F diving (4) dt dt π The damped oscillation is chaacteized by two time constants: the undamped peiod T ω0 and the amplitude elaxation timeτ. γ ω0 The Q mech facto of the oscillato is defined as: Q mech (4) γ Qmech Note that coesponds to the numbe of oscillations duing a decay of amplitude to /e of π its initial value. Appaatus notes Use the expeimental aangement fom Figue. Thee will be only one inducto coil povided. It will be used in Execises and as L in the LC cicuit, and in Execise 3 as pickup coil (it has to be mounted close to the tuning fok am). When setting up the oscilloscope, you may toggle the BW limit ON, to filte some of the noise. Connect Ch. to the function geneato output, using a Tee connecto. Execise : Fee decay Use L and a C of ~0000pF. You may use a G bidge fom the esouce Cente to accuately measue L and C. Calculate ω (the esonance fequency of the LC cicuit). Connect L-C in seies with ~000Ω and a signal geneato. Note that if the oscilloscope is connected acoss, cuent i can be monitoed. Estimate Q using Eq.. To obseve the fee decay, use a squae wave with a long peiod compaed to π/ω, and vaious values of, including 0. (3) - 4 -

5 Execise : Sine wave esponse Obseve the shift in phase of V c elative to the geneato voltage nea the esonance. Plot this phase shift vs. ω and also the magnitude of the cicuit impedance vs. ω, using log-log coodinates to locate the esonance and the half-powe fequency points. Use the half-powe points to ecalculate Q. Python equiement (PHY4/34 students only): do the plots mentioned above using a Python pogam. Output Q. Question: Why ae Q values lowe than the value calculated using Eq. ()? Optional: If L and C ae connected in paallel instead of seies the oles of i and v ae intechanged: the cuent is a minimum at esonance. This is called a "cuent tank". The analysis is complicated by the coil esistance still being in seies with L. It's not so simple to sot out the effective Q in this case unless it is vey lage. Ty it if you wish. Execise 3: The tuning fok In a low fequency cicuit using coils, it is nealy impossible to achieve Q > 50 because of the coil electic esistance. A mechanical system can do much bette: Q ~ 0 4 o moe is feasible. Set up the tuning fok without signal geneato. Connect it to the oscilloscope using the dive coil connectos. Note that the signal picked up is popotional to fok am velocity. Pinch the fok. Using the UN/STOP function, feeze the fee decay and measue the aveage peiod T π / ω0. In ode to get the amplitude elaxation time / γ, you have to switch the oscilloscope time base to seconds. Evaluate Q mech. Question: How can the dive coil pick up the fok oscillation? Connect the geneato to the dive coil and to Ch. of oscilloscope. Setup a sinusoidal wave in the ange 50-00Hz. Mount the pickup coil, connect it to Ch. and obtain the sinusoidal esponse of the fok (always allow fo the fact that some of the output is diect pickup fom the dive coil). Slowly otate the pickup coil it until the signal/noise atio is optimal. Caefully tune the fequency until you each the esonance. Please note that esonance is vey naow (occus within - Hz). Within small limits, the fok will tend to "pull" the geneato into the ight elation. This is a pimitive example of a "phase lock". Find the ½ powe points and calculate anothe value fo Q mech. Comment on diffeences and eo souces. Python equiement (PHY4/34 students only): Evaluate Q mech.and the ½ points as outputs of anothe Python pogam to fit the data fom Execise 3 Question: What is the ole of the magnets mounted at the end of fok ams? The long tem stability of the fok means it can be used to set the fequency of an othewise boad fequency system. This is the low fequency analogue to oscillatos which use piezoelectic cystals with MHz esonant fequencies. Optional:Ty diving the fok with the squae wave signal. The high Q ensues esponse only at cetain Fouie components of the squae wave. This expeiment was evised in 007 by uxanda Sebanescu and Luke Helt. evised in 009 by MS